Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

The non-stationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the non-stationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties which yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons with the data obtained by digital simulation show that the method, being non-perturbative in nature, yields reliable results even for large values of the nonlinearity parameter.